Journal of Geodetic Science • 2(4) • 2012 • 319-324 DOI: 10.2478/v10156-011-0046-3 • A conventional approach for comparing vertical reference frames Research Article C. Kotsakis∗ Department of Geodesy and Surveying School of Engineering Aristotle University of Thessaloniki University Box 440 Thessaloniki 541 24, Greece Abstract: A conventional transformation model between different realizations of a vertical reference system is an important tool for geodetic studies related to precise vertical positioning and physical height determination. Its fundamental role is the evaluation of the consistency for co- located vertical reference frames that are obtained from different observation techniques, data sources or optimal estimation strategies in terms of an appropriate set of “vertical datum perturbation” parameters. Our scope herein is to discuss a number of key issues related to the formulation of such a transformation model and to present some simple examples from its practical implementation in the comparison of existing vertical frames over Europe. Keywords: Conventional height transformation • physical heights • vertical datums • vertical reference frames © Versita sp. z o.o. Received 26-09-2012; accepted 30-10-2012 1. Introduction reference frames, and also for assisting their quality assessment through a suitable de-trending of their systematic differences in order to identify any localized distortions in their respective coor- dinate sets. The comparison of terrestrial reference frames (TRFs) that are ob- tained by different observation techniques, modeling assumptions A similar situation as the one described above occurs also in geode- and optimal estimation strategies is a common geodetic problem tic studies related to the establishment of vertical reference frames constituting either a research goal in itself or an auxiliary task for (VRFs) for physical height determination. Different realizations of a other geodetic applications. Such a comparison relies on the use vertical reference system (VRS) may be available over a regional or of the linearized similarity transformation, also known as Helmert even continental network, originating from separate leveling cam- transformation (e.g. Leick and van Gelder 1975), which supports paigns, alternative data sources and modeling strategies. As an ex- the evaluation of TRFs on the basis of datum-perturbation param- ample, consider a set of national leveling benchmarks that is part eters that are associated with the theoretical denition of a ter- of the United European Leveling Network (UELN): three vertical restrial reference system (Altamimi et al. 2007). Based on the frames co-exist in such a leveling network whose physical heights least squares adjustment of this model over a network of com- are respectively obtained from the EVRF00 and EVRF07 continen- mon stations, a set of estimated parameters is obtained that quan- tal solutions (Ihde and Augath 2001, Sacher et al. 2008) and also tify the origin, orientation and metric consistency of two TRFs by the (usually older) national adjustment of the primary height in terms of their relative translation, rotation and scale variation. network in the underlying country. If, in addition, Global Position- The aforementioned scheme provides a geodetically meaning- ing System (GPS) data are available at the particular stations, then ful framework for comparing and transforming Euclidean spatial more VRFs could emerge through the synergetic use of gravimet- ric geoid models that enable the conversion of observed geometric ∗E-mail: [email protected] heights to physical heights. 320 Journal of Geodetic Science An objective comparison of different VRFs needs to be based on meaning (if any) of its associated parameters. In fact, the estimated a conventional model that is able to map the differences of co- values of x have never been of any actual importance in geodetic located physical heights to a set of geodetically meaningful param- studies, other than offering a more or less arbitrary parametric de- ′ eters. The adopted model must resemble the role of the Helmert scription for the overall trend of the height differences Hi − Hi in transformation while its associated parameters should reect the support of GPS-based leveling techniques within an existing verti- vertical datum disturbance implied by the corresponding height cal datum. datasets. Eventually, the utmost role of such a model is to be used It is worth noting that the use of the well-known 4-parameter for generating a combined optimal VRF solution from multiple real- model (Heiskanen and Moritz 1967, ch. 5) izations that are jointly merged into a unied vertical frame by pos- T tulating appropriate minimum constraints to the datum-related ai x = x0+∆x cos ϕi cos λi+∆y cos ϕi sin λi+∆z sin ϕi (2) parameters of the underlying height transformation model. The aim of this paper is to discuss some general aspects about the may be viewed, to some extent, as an attempt to infer hidden “da- ′ formulation of a conventional height transformation model and to tum disturbances” between the height frames {Hi} and {Hi }. present a few examples from its practical use in the comparison of Such a viewpoint relies on the equivalent form of Eq. (1) existing VRFs over Europe. ′ T Ni − Ni = ai x + si + vi i = 1, 2, ..., m (3) 2. Height transformation schemes in practice ′ Several transformation schemes for physical heights are commonly where Ni and Ni denote the geoid undulations obtained from a used in geodetic practice. Typical examples include the reduc- gravimetric model and GPS/leveling data, respectively. In view of tion of physical heights to a conventional permanent tide sys- Eq. (3), the use of the 4-parameter model with Eq. (1) implies that ′ tem and/or to a reference time epoch due to temporal variations the systematic part of the differences Hi − Hi is essentially de- caused by various geodynamical effects (Ekman 1989; Mäkinen scribed through a 3D spatial shift (∆x, ∆y, ∆z) and an apparent and Ihde 2009), the conversion from normal to orthometric heights scale change (x0) between the corresponding reference surfaces and vice versa (Flury and Rummel 2009; Sjöberg 2010), and the of the physical heights (see Kotsakis 2008). determination of apparent height variations due to a geopoten- The aforementioned 4-parameter model was regularly used in tial offset in the zero-height level of the underlying vertical datum older studies for estimating geodetic datum differences from het- (Jekeli 2000). Also a number of empirical transformation schemes erogeneous height data; especially for assessing the geocentric- have been employed for the analysis of co-located heterogeneous ity of TRFs based on Doppler-derived and gravimetrically-derived heights and the inference of systematic differences between them. geoid undulations and also for determining the Earth’s optimal A classic example is the combined adjustment of ellipsoidal, geoid equatorial radius from geometric and physical heights (e.g. Schaab and leveled height data which can be perceived as a parameter es- and Groten 1979, Grappo 1980, Soler and van Gelder 1987). These timation problem in a generalized height transformation model: tasks require a global height data distribution otherwise the trans- lation parameters ∆x, ∆y, ∆z and the scaling term x0 become ′ T highly correlated, and their adjusted values may be totally unreal- Hi − Hi = ai x + si + vi i = 1, 2, . , m (1) istic from a physical point of view. This is the reason that the LS in- ′ The terms Hi and Hi correspond to the orthometric heights ob- version of Eq. (1) will not always produce a geodetically meaning- tained from leveling measurements and GPS/geoid data, respec- ful solution for the individual parameters of the 4-parameter model tively. Their systematic differences are usually modeled by a low- (not even for the estimated “height bias” x0) when applied over a T order parametric component ai x and (optionally) a spatially corre- regional test network; for some numerical examples see Kotsakis lated zero-mean signal si, whereas vi contains the remaining ran- and Katsambalos (2010). Moreover, a theoretical drawback of this dom errors in the height data. The estimated values of the model model for VRF comparison studies is that it neglects one of the key parameters and the predicted values of the stochastic signals can parameters for the denition and realization of vertical datums: a be jointly obtained from the least squares (LS) inversion of Eq. (1) geopotential reference value W0 and, more importantly, its actual using some a-priori information for the data noise level and the sig- and/or apparent variation between different VRFs. nal covariance function; for more details see Kotsakis and Sideris As a closing remark, we should note that the comparison of co- (1999). located vertical frames needs to consider their scale variation due Numerous modelling options have been followed in practice for to systematic differences originating from the measurement tech- T the parametric term ai x in Eq. (1), none of which has ever served niques and data modeling options that were used for the determi- as a “geodetically meaningful” transformation model between the nation of the physical heights in each frame. In fact, one should not underlying VRFs - that is, between the leveling-based frame {Hi} forget that the fundamental theoretical constraint h−H −N = 0 ′ and the GPS/geoid-based frame Hi . In most cases, the suitability of requires not only the “origin consistency” among the heteroge- the